The article presents the results of mathematical and computer modeling of the oscillatory processes of viscoelastic pipelines considering stationary external loads. A mathematical model of viscoelastic pipeline vibrations based on the theory of beams was developed when a pulsating fluid flows through it. Using the Bubnov-Galerkin method, based on a polynomial approximation of deflections, the problem was reduced to the study of a system of ordinary integro-differential equations, the solution of which was found numerically. A computational algorithm has been developed to solve vibration problems of composite pipelines conveying pulsating fluid. Stability and amplitude-time characteristics of vibrations of composite pipelines conveying pulsating fluid were studied at wide range of parameters variation of deformable systems and fluid flow. Critical fluid flow rates were found at which a viscoelastic pipe lost its rectilinear equilibrium shape. A singularity effect in hereditary kernels on vibrations of structures with viscoelastic properties was numerically studied. It was shown that with an increase in viscosity parameter of the pipeline material, the critical flow rate decreases. It was revealed that an increase in fluid pulsation frequency and excitation coefficient led to a decrease in the fluid flow critical rate. It was stated that an increase in Winkler bases parameters and the rigidity parameter of a continuous layer led to an increase in the critical flow rate.